Towards a characterization of matrices admitting wavelet sets

Let A be an invertible N × N matrix, and let Γ be a full-rank lattice in RN. An (A, Γ)
orthonormal wavelet is a function ψ ∈ L2(RN ) such that

{|A|j/2ψ(Aj x + k) : j ∈ Z, k ∈ Γ}

is an orthonormal basis for L2(RN). There is no characterization of pairs (A,Γ) for which
there exists an orthonormal wavelet. In this talk, we consider an ostensibly simpler problem
of characterizing the pairs (A, Γ) for which there exists a wavelet set; that is, a set W ⊂ RN
such that the indicator function on W is the Fourier transform of an (A,Γ) orthonormal
wavelet. Equivalently, we wish to characterize pairs (A,Γ) such that there exists W with
{Aj(W):j∈Z} and {W+k:k∈Γ} are measurable tilings of RN. I will reporton the
state of the problem and give some modest, new partial results.